These three numbers are not three digits; they are numbers made of digits. (e.g. "21" has two digits just as "480" has three digits.)

A digit is a numeric symbol (such as "2" or "5") used in combinations (such as "25") to represent numbers (such as the number 25) in positionalnumeral systems. The name "digit" comes from the fact that the ten digits (Latindigiti meaning fingers)[1] of the hands correspond to the ten symbols of the common base 10numeral system, i.e. the decimal (ancient Latin adjective decem meaning ten)[2] digits.

In a given numeral system, if the base is an integer, the number of digits required will always be equal to the absolute value of the base, for example, the decimal system (base 10) has ten digits (0 through to 9), whereas binary (base 2) has two digits (0 and 1).

In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length, each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.

Each digit in a number system represents an integer, for example, in decimal the digit "1" represents the integer one, and in the hexadecimal system, the letter "A" represents the number ten. A positional number system must have a digit representing the integers from zero up to, but not including, the radix of the number system.

Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position, the number 12 can be expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 can be expressed by three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.

The Hindu–Arabic numeral system (or the Hindu numeral system) uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the "ones place" or "units place",[3][clarification needed] which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base, for example, in the numeral 10.34 (written in base 10),

the 0 is immediately to the left of the separator, so it is in the ones or units place, and is called the units digit or ones digit[4][clarification needed];

the 1 to the left of the ones place is in the tens place, and is called the tens digit;

the 3 is to the right of the ones place, so it is in the tenths place, and is called the tenths digit;

the 4 to the right of the tenths place is in the hundredths place, and is called the hundredths digit.

The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place.

The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems, the calculation involves the multiplication of the given digit by the base raised by the exponent n − 1, where n represents the position of the digit from the separator; the value of n is positive (+), but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative (−) n, for example, in the number 10.34 (written in base 10),

the 1 is second to the left of the separator, so based on calculation, its value is,

n−1=2−1=1{\displaystyle n-1=2-1=1}

1×101=10{\displaystyle 1\times 10^{1}=10}

the 4 is second to the right of the separator, so based on calculation its value is,

The first true written positional numeral system is considered to be the Hindu–Arabic numeral system, this system was established by the 7th century in India,[5] but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.[5]

The digits of the Maya numeral system, with Hindu–Arabic equivalents

By the 13th century, Hindu–Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). They began to enter common use in the 15th century. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system, the system was vigesimal (base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom, the Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.

The rod numerals, the written forms of counting rods once used by Chinese and Japanese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system, the Suzhou numerals are variants of rod numerals.

The binary (base 2), octal (base 8), and hexadecimal (base 16) systems, extensively used in computer science, all follow the conventions of the Hindu–Arabic numeral system. The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7", the hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively.

Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics. Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.

Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let f(x){\displaystyle f(x)} represent the digital root of x{\displaystyle x}, as described above. Casting out nines makes use of the fact that if A+B=C{\displaystyle A+B=C}, then f(f(A)+f(B))=f(C){\displaystyle f(f(A)+f(B))=f(C)}. In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal the original addition must have been faulty.

Repunits are integers that are represented with only the digit 1, for example, 1111 (one thousand, one hundred and eleven) is a repunit. Repdigits are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit, the primacy of repunits is of interest to mathematicians.[6]

Palindromic numbers are numbers that read the same when their digits are reversed. A Lychrel number is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed, the question of whether there are any Lychrel numbers in base 10 is an open problem in recreational mathematics; the smallest candidate is 196.

Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers, the Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.

To preserve numerical information, tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods.

A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC, written numbers were dissociated from the things being counted and became abstract numerals.

Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay, these cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations, this system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees).

In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add, this makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide; in modern times modular arithmetic is sometimes used in digital signal processing.

The oldest Greek system was that of the Attic numerals, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals). Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers, the Roman numerals system remained in common use in Europe until positional notation came into common use in the 16th century.

The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.[7] They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.

The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanishconquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

Some authorities believe that positional arithmetic began with the wide use of counting rods in China, the earliest written positional records seem to be rod calculus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932.[citation needed]

From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.[citation needed]

The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching from China. Binary numbers came into common use in the 20th century because of computer applications.[citation needed]

Arabic numerals
–
In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to su

Number
–
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculation

1.
The number 605 in Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.

Positional notation
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can

Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers

1.
Numeral systems

Latin
–
Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages

Base 10
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal repr

1.
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.

3.
Ten fingers on two hands, the possible starting point of the decimal counting.

4.
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period

Radix
–
In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for

Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur

1.
The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

Decimal
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal repr

1.
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.

3.
Ten fingers on two hands, the possible starting point of the decimal counting.

4.
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period

One
–
1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the Eng

Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a

10 (number)
–
10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the a

Positional number system
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can

Zero
–
Gonzalo Barrios, known by his gamertag ZeRo, is a Chilean professional Super Smash Bros. player. He is considered one of the best Super Smash Bros. for Wii U players in the world, ZeRo had a record-breaking 56-tournament winning streak in 2015, in which he also won several high-profile tournaments like EVO2015 and The Big House 5. In the past has a

Decimal separator
–
A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark, the choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is a

Europe
–
Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers about 10,180,000 square kilometres, or 2% of the Earths surface, politically, Europ

3.
A medieval T and O map from 1472 showing the three continents as domains of the sons of Noah — Asia to Sem (Shem), Europe to Iafeth (Japheth), and Africa to Cham (Ham)

4.
Early modern depiction of Europa regina ('Queen Europe') and the mythical Europa of the 8th century BC.

Positional numeral system
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can

Glyph
–
In typography, a glyph /ˈɡlɪf/ is an elemental symbol within an agreed set of symbols, intended to represent a readable character for the purposes of writing. In Turkish, however, it is not a glyph because that language has two versions of the letter i, with and without a dot. In Japanese syllabaries, a number of the characters are made up of more

Fibonacci
–
Fibonacci was an Italian mathematician, considered to be the most talented Western mathematician of the Middle Ages. The name he is called, Fibonacci, is short for figlio di Bonacci and he is also known as Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo. Fibonacci popularized the Hindu–Arabic numeral system in the Western World primaril

1.
Portrait by an unknown artist

2.
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.

3.
19th century statue of Fibonacci in Camposanto, Pisa.

Liber Abaci
–
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of

Maya numerals
–
The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of tw

3.
Detail showing three columns of glyphs from La Mojarra Stela 1. The left column uses Maya numerals to show a Long Count date of 8.5.16.9.7, or 156 CE.

Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters

Mayas
–
The Maya people are a group of Indigenous peoples of Mesoamerica. They inhabit southern Mexico, Guatemala, Belize, El Salvador, the pre-Columbian Maya population was approximately eight million. There were a seven million Maya living in this area at the start of the 21st century. Guatemala, southern Mexico and the Yucatán Peninsula, Belize, El Salv

Thai numerals
–
The Thai language lacks grammatical number. A count is expressed in the form of an uninflected noun followed by a number. In Thai, counting is kannap, the classifier, laksananam Variations to this pattern do occur, a partial list of Thai words that also classify nouns can be found in Wiktionary category, Thai classifiers. Thai sūn is written as ova

Thailand
–
Thailand, officially the Kingdom of Thailand, formerly known as Siam, is a country at the centre of the Indochinese peninsula in Southeast Asia. With a total area of approximately 513,000 km2, Thailand is the worlds 51st-largest country and it is the 20th-most-populous country in the world, with around 66 million people. The capital and largest cit

Counting rods
–
Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient China, Japan, Korea, and Vietnam. They are placed horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals and they are a true positional numeral system with digits

China
–
China, officially the Peoples Republic of China, is a unitary sovereign state in East Asia and the worlds most populous country, with a population of over 1.381 billion. The state is governed by the Communist Party of China and its capital is Beijing, the countrys major urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin

Japan
–
Japan is a sovereign island nation in Eastern Asia. Located in the Pacific Ocean, it lies off the eastern coast of the Asia Mainland and stretches from the Sea of Okhotsk in the north to the East China Sea, the kanji that make up Japans name mean sun origin. 日 can be read as ni and means sun while 本 can be read as hon, or pon, Japan is often referr

Chinese numerals
–
Chinese numerals are words and characters used to denote numbers in Chinese. Today speakers of Chinese use three written numeral systems, the system of Arabic numerals used worldwide, and two indigenous systems, the more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared wi

2.
Chinese and Arabic numerals may coexist, as on this kilometer marker: 1620 km on Hwy G209 (G二〇九)

3.
Counting rod numerals

4.
Shang oracle bone numerals of 14th century B.C.

Binary numeral system
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers

Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decima

Computer science
–
Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems an

3.
The German military used the Enigma machine (shown here) during World War II for communications they wanted kept secret. The large-scale decryption of Enigma traffic at Bletchley Park was an important factor that contributed to Allied victory in WWII.

Ternary numeral system
–
The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less com

Balanced ternary
–
Balanced ternary is a non-standard positional numeral system, useful for comparison logic. While it is a number system, in the standard ternary system. The digits in the balanced ternary system have values −1,0, different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T represents −1, while 0 a

Setun
–
Setun was a computer developed in 1958 at Moscow State University. It was built under the leadership of Sergei Sobolev and Nikolay Brusentsov and it was the most modern ternary computer, using the balanced ternary numeral system and three-valued ternary logic instead of the two-valued binary logic prevalent in other computers. The computer was buil

Augustin-Louis Cauchy
–
Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence ove

1.
Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.

2.
The title page of a textbook by Cauchy.

3.
Leçons sur le calcul différentiel, 1829

Signed-digit representation
–
In mathematical notation for numbers, signed-digit representation is a positional system with signed digits, the representation may not be unique. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries, in the binary numeral system, a special case signed-digit representat

Florian Cajori
–
Florian Cajori was a Swiss-American historian of mathematics. Florian Cajori immigrated to the United States at the age of sixteen and he received both his bachelors and masters degrees from the University of Wisconsin–Madison. He taught for a few years at Tulane University, before being appointed as professor of applied mathematics there in 1887 a

Computer design
–
In computer engineering, computer architecture is a set of rules and methods that describe the functionality, organization, and implementation of computer systems. Some definitions of architecture define it as describing the capabilities and programming model of a computer, in other definitions computer architecture involves instruction set archite

Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical stru

Equation
–
In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, iden

Prime number
–
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifyi

1.
The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.

History of writing ancient numbers
–
The first numbers were invented in India. The first method of counting was counting on fingers and this evolved into sign language for the hand-to-eye communication of numbers which, while not writing, gave way to written numbers. Tallies made by carving notches in wood, bone, and stone were used for at least forty thousand years, stone age culture

Tally marks
–
Tally marks, also called hash marks, are a unary numeral system. They are a form of used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, however, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were histori

2.
In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.

Indigenous peoples of the Americas
–
The indigenous peoples of the Americas are the pre-Columbian peoples of the Americas and their descendants. The term Amerindian is used in Quebec, the Guianas, Indigenous peoples of the United States are commonly known as Native Americans or American Indians, and Alaska Natives. Application of the term Indian originated with Christopher Columbus, w

Sumer
–
Living along the valleys of the Tigris and Euphrates, Sumerian farmers were able to grow an abundance of grain and other crops, the surplus of which enabled them to settle in one place. Proto-writing in the dates back to c.3000 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr and date back to 3300 BC, modern historians have sugge

Mixed radix
–
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the smaller one. 32,5,7,45,15,500. ∞,7,24,60,60,1000 or as 32∞577244560.15605001000 In the tabu

Time
–
Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, nevertheless,

1.
The flow of sand in an hourglass can be used to keep track of elapsed time. It also concretely represents the present as being between the past and the future.

Angle
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed

1.
An angle enclosed by rays emanating from a vertex.

Modular arithmetic
–
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is div

2.
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.

4.
An example of handwritten Bengali. Part of a poem written in Bengali (and with its English translation below each Bengali paragraph) by Nobel Laureate Rabindranath Tagore in 1926 in Hungary.

LIST OF IMAGES

1.
Arabic numerals
–
In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to suggest that the numerals in their current form developed from Arabic letters in the Maghreb, the current form of the numerals developed in North Africa, distinct in form from the Indian and eastern Arabic numerals. The use of Arabic numerals spread around the world through European trade, books, the term Arabic numerals is ambiguous. It most commonly refers to the widely used in Europe. Arabic numerals is also the name for the entire family of related numerals of Arabic. It may also be intended to mean the numerals used by Arabs and it would be more appropriate to refer to the Arabic numeral system, where the value of a digit in a number depends on its position. The decimal Hindu–Arabic numeral system was developed in India by AD700, the development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmaguptas formulation of zero as a number in AD628. The system was revolutionary by including zero in positional notation, thereby limiting the number of digits to ten. It is considered an important milestone in the development of mathematics, one may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0123456789. The first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the symbol for zero in them, dated back as far as the 6th century AD. Inscriptions in Indonesia and Cambodia dating to AD683 have also been found and their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. Ghubar numerals themselves are probably of Roman origin, some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, Algoritmi, the translators rendition of the authors name, gave rise to the word algorithm

Arabic numerals
–
Numeral systems
Arabic numerals
–
Modern-day Arab telephone keypad with two forms of Arabic numerals: Western Arabic/European numerals on the left and Eastern Arabic numerals on the right
Arabic numerals
–
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Arabic numerals
–
Woodcut showing the 16th century astronomical clock of Uppsala Cathedral, with two clockfaces, one with Arabic and one with Roman numerals.

2.
Number
–
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not

3.
Positional notation
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100

4.
Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea

Numeral system
–
Numeral systems

5.
Latin
–
Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole

6.
Base 10
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0

Base 10
–
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.
Base 10
–
Numeral systems
Base 10
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Base 10
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period

7.
Radix
–
In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base

8.
Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number

Absolute value
–
The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

9.
Decimal
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0

Decimal
–
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.
Decimal
–
Numeral systems
Decimal
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period

10.
One
–
1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few

One
–
The 24-hour tower clock in Venice, using J as a symbol for 1.
One
–
"One" and "No. 1" redirect here. For other uses, see One (disambiguation).

11.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus &#x2019, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly

12.
10 (number)
–
10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十

13.
Positional number system
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100

14.
Zero
–
Gonzalo Barrios, known by his gamertag ZeRo, is a Chilean professional Super Smash Bros. player. He is considered one of the best Super Smash Bros. for Wii U players in the world, ZeRo had a record-breaking 56-tournament winning streak in 2015, in which he also won several high-profile tournaments like EVO2015 and The Big House 5. In the past has also been a top ranked Super Smash Bros and he mains Diddy Kong in Super Smash Bros. for Wii U, and mained Pit in Project M, Meta Knight in Brawl, and Fox in Melee. Barrios has had ZeRo as his gamertag since 2005 and he has been playing Smash since Super Smash Bros. in 1999. He started to travel and play in Melee tournaments in a local Akiba Game Store in early 2007, ZeRo quit Smash completely until December 2010 and then focused only on Brawl. ZeRo placed second in Brawl at Apex 2014, losing to Nairo and was the champion of the Smash Wii U at Apex 2015 and he defeated Dabuz, who was playing Captain Olimar, in the finals. ZeRo qualified for the MLG Anaheim 2014 championship bracket and finished 17th, ZeRo was ranked in 2014 by Melee it on Me as the 35th best Melee player in the world. On November 25,2014, he criticized Diddy Kongs repetitive playstyle in Smash Wii U, however, ZeRo later retracted this statement, and now says Diddy Kong is his favorite character to play with. Barrios streams on twitch. tv Mondays through Thursdays at 1PM PST, ZeRo attributes much of his success to training with Mew2King. He was sponsored by CLASH Tournaments for part of 2014 until resigning in November, e-Sports Earnings estimates that ZeRo has earned a career total of US$31,484.22 from tournaments. ZeRo was considered the third best Brawl player in the world by CLASH Tournaments in the 2014 SSBBRank, in early 2014, he picked up Melee again, as well as the mod Project M. Since November 2014 when he placed third at Skys Smash 4 Invitational, ZeRo won EVO2015, the largest Smash for Wii U. tournament at the time beating Mr. R in the finals. On the August 1,2015, Team SoloMid announced ZeRo as the player in their Super Smash Bros. division. At The Big House 5, ZeRo was knocked into Losers bracket very early and he qualified for the top 32. He led off by defeating Sonic main StaticManny, Sheik main top Melee Captain Falcon main Wizzrobe, Mario main Ally, in top 8, he beat Mario main 2Scoops Zenyou, Rosalina and Luma main Raquayza07, and beat Mario main ANTi in a very close set. In Losers finals, he handily beat Dabuz, who now plays Rosalina and Luma, after easily beating Nairo the first set of Grand Finals, ZeRo clutched out a win in the second set, 3-2, to win TBH5, despite suffering an early upset. In MLG World Finals 2015, ZeRo defeated Ness main Nakat, StaticManny, Ally, there, Nairo took two sets off of ZeRo, ending ZeRos reign at 56 tournaments. Kotaku named ZeRo, The Smash Bros, champ as one of the gamers of 2015, honoring his 56 tournament streak

Zero
Zero
Zero

15.
Decimal separator
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A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark, the choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article. In mathematics the decimal mark is a type of radix point, in the Middle Ages, before printing, a bar over the units digit was used to separate the integral part of a number from its fractional part, e. g.9995. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear, a similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e. g.9995. Later, a separatrix between the units and tenths position became the norm among Arab mathematicians, e. g. 99ˌ95, when this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used in England as an L-shaped or vertical bar before the popularization of the period, gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century, in France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position and it has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits, in some countries, a raised dot or dash may be used for grouping or decimal mark, this is particularly common in handwriting. In the United States, the stop or period was used as the standard decimal mark. g. However, as the mid dot was already in use in the mathematics world to indicate multiplication. In the event, the point was decided on by the Ministry of Technology in 1968, the three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua, all use the comma as the decimal mark. Interlingua has used the comma as its decimal mark since the publication of the Interlingua Grammar in 1951, Esperanto also uses the comma as its official decimal mark, while thousands are separated by non-breaking spaces,12345678,9. Idos Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido officially states that commas are used for the mark while full stops are used to separate thousands, millions. So the number 12,345,678.90123 for instance, the 1931 grammar of Volapük by Arie de Jong uses the comma as its decimal mark, and uses the middle dot as the thousands separator. In 1958, disputes between European and American delegates over the representation of the decimal mark nearly stalled the development of the ALGOL computer programming language. ALGOL ended up allowing different decimal marks, but most computer languages, the 22nd General Conference on Weights and Measures declared in 2003 that the symbol for the decimal marker shall be either the point on the line or the comma on the line. It further reaffirmed that numbers may be divided in groups of three in order to facilitate reading, neither dots nor commas are ever inserted in the spaces between groups

16.
Europe
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Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers about 10,180,000 square kilometres, or 2% of the Earths surface, politically, Europe is divided into about fifty sovereign states of which the Russian Federation is the largest and most populous, spanning 39% of the continent and comprising 15% of its population. Europe had a population of about 740 million as of 2015. Further from the sea, seasonal differences are more noticeable than close to the coast, Europe, in particular ancient Greece, was the birthplace of Western civilization. The fall of the Western Roman Empire, during the period, marked the end of ancient history. Renaissance humanism, exploration, art, and science led to the modern era, from the Age of Discovery onwards, Europe played a predominant role in global affairs. Between the 16th and 20th centuries, European powers controlled at times the Americas, most of Africa, Oceania. The Industrial Revolution, which began in Great Britain at the end of the 18th century, gave rise to economic, cultural, and social change in Western Europe. During the Cold War, Europe was divided along the Iron Curtain between NATO in the west and the Warsaw Pact in the east, until the revolutions of 1989 and fall of the Berlin Wall. In 1955, the Council of Europe was formed following a speech by Sir Winston Churchill and it includes all states except for Belarus, Kazakhstan and Vatican City. Further European integration by some states led to the formation of the European Union, the EU originated in Western Europe but has been expanding eastward since the fall of the Soviet Union in 1991. The European Anthem is Ode to Joy and states celebrate peace, in classical Greek mythology, Europa is the name of either a Phoenician princess or of a queen of Crete. The name contains the elements εὐρύς, wide, broad and ὤψ eye, broad has been an epithet of Earth herself in the reconstructed Proto-Indo-European religion and the poetry devoted to it. For the second part also the divine attributes of grey-eyed Athena or ox-eyed Hera. The same naming motive according to cartographic convention appears in Greek Ανατολή, Martin Litchfield West stated that phonologically, the match between Europas name and any form of the Semitic word is very poor. Next to these there is also a Proto-Indo-European root *h1regʷos, meaning darkness. Most major world languages use words derived from Eurṓpē or Europa to refer to the continent, in some Turkic languages the originally Persian name Frangistan is used casually in referring to much of Europe, besides official names such as Avrupa or Evropa

Europe
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Reconstruction of Herodotus ' world map
Europe
Europe
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A medieval T and O map from 1472 showing the three continents as domains of the sons of Noah — Asia to Sem (Shem), Europe to Iafeth (Japheth), and Africa to Cham (Ham)
Europe
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Early modern depiction of Europa regina ('Queen Europe') and the mythical Europa of the 8th century BC.

17.
Positional numeral system
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100

18.
Glyph
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In typography, a glyph /ˈɡlɪf/ is an elemental symbol within an agreed set of symbols, intended to represent a readable character for the purposes of writing. In Turkish, however, it is not a glyph because that language has two versions of the letter i, with and without a dot. In Japanese syllabaries, a number of the characters are made up of more than one separate mark, however, in some cases, additional marks fulfill the role of diacritics, to differentiate distinct characters. In general, a diacritic is a glyph, even if it is contiguous with the rest of the character, two or more glyphs which have the same significance, whether used interchangeably or chosen depending on context, are called allographs of each other. The term has been used in English since 1727, borrowed from glyphe, from the Greek γλυφή, glyphē, carving, and the verb γλύφειν, glýphein, to hollow out, engrave, carve. The word glyph first came to widespread European attention with the engravings, in archaeology, a glyph is a carved or inscribed symbol. It may be a pictogram or ideogram, or part of a system such as a syllable. In 1897 Dana Evans discovered glyphs written on rocks in the Colorado Desert and these ancient characters have been called the most enlightening discovery in Native American History in the 19th Century. In typography, a glyph has a different definition, it is the specific shape, design. The same is true in computing, in computing as well as typography, the term character refers to a grapheme or grapheme-like unit of text, as found in natural language writing systems. The range of glyphs required increases correspondingly, in summary, in typography and computing, a glyph is a graphical unit. In graphonomics, the glyph is used for a noncharacter. Most typographic glyphs originate from the characters of a typeface, in the mobile text input technologies, Glyph is a family of text input methods based on the decomposition of letters into basic shapes. In role-playing games, the glyph is sometimes used alongside the word rune in describing magical drawings or etchings. Runes often refer to placing the image on an object or person to empower it, whereas the magic in a glyph lies dormant and is only triggered when the glyph is read or approached

19.
Fibonacci
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Fibonacci was an Italian mathematician, considered to be the most talented Western mathematician of the Middle Ages. The name he is called, Fibonacci, is short for figlio di Bonacci and he is also known as Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo. Fibonacci popularized the Hindu–Arabic numeral system in the Western World primarily through his composition in 1202 of Liber Abaci and he also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci was born around 1175 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, Guglielmo directed a trading post in Bugia, a port in the Almohad dynastys sultanate in North Africa. Fibonacci travelled with him as a boy, and it was in Bugia that he learned about the Hindu–Arabic numeral system. Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and he soon realised the many advantages of the Hindu-Arabic system. In 1202, he completed the Liber Abaci which popularized Hindu–Arabic numerals in Europe, Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. The date of Fibonaccis death is not known, but it has estimated to be between 1240 and 1250, most likely in Pisa. In the Liber Abaci, Fibonacci introduced the so-called modus Indorum, the book advocated numeration with the digits 0–9 and place value. The book was well-received throughout educated Europe and had a impact on European thought. No copies of the 1202 edition are known to exist, the book also discusses irrational numbers and prime numbers. Liber Abaci posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions, the solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonaccis Liber Abaci contains the earliest known description of the sequence outside of India, in the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0,1,1,2, as modern mathematicians do but with 1,1,2, etc. He carried the calculation up to the place, that is 233. Fibonacci did not speak about the ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was constructed and raised in Pisa, today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers, examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period

Fibonacci
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Portrait by an unknown artist
Fibonacci
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A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.
Fibonacci
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19th century statue of Fibonacci in Camposanto, Pisa.

20.
Liber Abaci
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Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the Hindu–Arabic numeral system. The title of Liber Abaci means The Book of Calculation, the second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found, the first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. The second section presents examples from commerce, such as conversions of currency and measurements, another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots, the book also includes proofs in Euclidean geometry. Fibonaccis method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam, there are three key differences between Fibonaccis notation and modern fraction notation. We generally write a fraction to the right of the number to which it is added. Fibonacci instead would write the same fraction to the left, i. e.132. That is, b a d c = a c + b c d, the notation was read from right to left. For example, 29/30 could be written as 124235 and this can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 14132 would represent the number that would now more commonly be written as the mixed number 2712, or simply the improper fraction 3112. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the break in the bar. If all numerators are 1 in a written in this form, and all denominators are different from each other. This notation was also combined with the composite fraction notation. The complexity of this notation allows numbers to be written in different ways. In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals

21.
Maya numerals
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The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty, for example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents one twenty or 1×20, which is added to three dots and two bars, or thirteen, upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots, the powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures, the face glyph for a number represents the deity associated with the number. These face number glyphs were used, and are mostly seen on some of the most elaborate monumental carving. Addition and subtraction, Adding and subtracting numbers below 20 using Maya numerals is very simple, addition is performed by combining the numeric symbols at each level, If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed, similarly with subtraction, remove the elements of the subtrahend symbol from the minuend symbol, If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column, the Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder within its vigesimal positional numeral system. A shell glyph – – was used as a symbol for these Long Count dates. However, since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero predated the Maya, indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, in the Long Count portion of the Maya calendar, a variation on the strictly vigesimal numbering is used. The Long Count changes in the place value, it is not 20×20 =400, as would otherwise be expected. This is supposed to be because 360 is roughly the number of days in a year, subsequent place values return to base-twenty. In fact, every known example of large numbers uses this modified vigesimal system and it is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system. Maya Mathematics - online converter from decimal numeration to Maya numeral notation, anthropomorphic Maya numbers - online story of number representations

22.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power

23.
Mayas
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The Maya people are a group of Indigenous peoples of Mesoamerica. They inhabit southern Mexico, Guatemala, Belize, El Salvador, the pre-Columbian Maya population was approximately eight million. There were a seven million Maya living in this area at the start of the 21st century. Guatemala, southern Mexico and the Yucatán Peninsula, Belize, El Salvador, one of the largest groups of modern Maya can be found in Mexicos Yucatán State and the neighboring states of Campeche, Quintana Roo and in Belize. These peoples commonly identify themselves simply as Maya with no further ethnic subdivision and they speak the language which anthropologists term Yucatec Maya, but is identified by speakers and Yucatecos simply as Maya. Among Maya speakers, Spanish is commonly spoken as a second or first language, linguists refer to the Maya language as Yucatec or Yucatec Maya to distinguish it from other Mayan languages. This norm has often been misinterpreted to mean that the people are also called Yucatec Maya, that refers only to the language. Maya is one language in the Mayan language family, thus, to refer to Maya as Mayans would be similar to referring to Spanish people as Romantics because they speak a language belonging to the Romance language family. Confusion of the term Maya/Mayan as an ethnic label occurs because Maya women who use traditional dress identify by the ethnic term mestiza, the Yucatáns indigenous population was first exposed to Europeans after a party of Spanish shipwreck survivors came ashore in 1511. One of the sailors, Gonzalo Guerrero, is reported to have taken up with a woman and started a family. Later Spanish expeditions to the region were led by Córdoba in 1517, Grijalva in 1518, from 1528 to 1540, several attempts by Francisco Montejo to conquer the Yucatán failed. His son, Francisco de Montejo the Younger, fared almost as badly when he first took over, while holding out at Chichen Itza. Chichen Itza was conquered by 1570, in 1542, the western Yucatán Peninsula also surrendered to him. Historically, the population in the half of the peninsula was less affected by. In the 21st century in the Yucatán Peninsula, between 750,000 and 1,200,000 people speak Mayan, however, three times more than that are of Maya origins, hold ancient Maya surnames, and do not speak Mayan languages as their first language. Matthew Restall, in his book The Maya Conquistador, mentions a series of letters sent to the King of Spain in the 16th and 17th centuries. The noble Maya families at that time signed documents to the Spanish Royal Family, surnames mentioned in letters are Pech, Camal, Xiu, Ucan, Canul, Cocom. A large 19th-century revolt by the native Maya people of Yucatán, for a period the Maya state of Chan Santa Cruz was recognized as an independent nation by the British Empire, particularly in terms of trading with British Honduras

24.
Thai numerals
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The Thai language lacks grammatical number. A count is expressed in the form of an uninflected noun followed by a number. In Thai, counting is kannap, the classifier, laksananam Variations to this pattern do occur, a partial list of Thai words that also classify nouns can be found in Wiktionary category, Thai classifiers. Thai sūn is written as oval 0 when using Arabic numerals, but a small circle ๐ when using traditional numerals and it is from Sanskrit śūnya, as are the alternate names for numbers one to four given below, but not the counting 1. Thai names for N +1 and the regular digits 2 through 9 as shown in the table, below, resemble those in Chinese varieties as spoken in Southern China, Thai and Lao words for numerals are almost identical, however, the numerical digits vary somewhat in shape. Shown below is a comparison between three languages using Cantonese and Minnan characters and pronunciations, the Thai transliteration uses the Royal Thai General System of Transcription. Sanskrit lakh designates the place value of a digit, which are named for the powers of ten, the place is lak nuai, tens place, lak sip, hundreds place, lak roi. The number one following any multiple of sip becomes et, the number ten is the same as Minnan 十. Numbers from twenty to twenty nine begin with yi sip, names of the lak sip for 30 to 90, and for the lak of 100,1000,10,000,100,000 and million, are almost identical to those of the like Khmer numerals. For the numbers twenty-one through twenty-nine, the part signifying twenty, yi sip, see the alternate numbers section below. The hundreds are formed by combining roi with the tens and ones values, for example, two hundred and thirty-two is song roi sam sip song. The words roi, phan, muen, and saen should occur with a preceding numeral, nueng never precedes sip, so song roi nueng sip is incorrect. Native speakers will sometimes use roi nueng with different tones on nueng to distinguish one hundred from one hundred, however, such distinction is often not made, and ambiguity may follow. To resolve this problem, if the number 101 is intended, numbers above a million are constructed by prefixing lan with a multiplier. For example, ten million is sip lan, and a trillion is lan lan, colloquially, decimal numbers are formed by saying chut where the decimal separator is located. For example,1.01 is nueng chut sun nueng, fractional numbers are formed by placing nai between the numerator and denominator or using x suan y to clearly indicate. For example, ⅓ is nueng nai sam or nueng suan sam, the word set can be omitted. The word khrueng is used for half and it precedes the measure word if used alone, but it follows the measure word when used with another number

25.
Thailand
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Thailand, officially the Kingdom of Thailand, formerly known as Siam, is a country at the centre of the Indochinese peninsula in Southeast Asia. With a total area of approximately 513,000 km2, Thailand is the worlds 51st-largest country and it is the 20th-most-populous country in the world, with around 66 million people. The capital and largest city is Bangkok, Thailand is a constitutional monarchy and has switched between parliamentary democracy and military junta for decades, the latest coup being in May 2014 by the National Council for Peace and Order. Its capital and most populous city is Bangkok and its maritime boundaries include Vietnam in the Gulf of Thailand to the southeast, and Indonesia and India on the Andaman Sea to the southwest. The Thai economy is the worlds 20th largest by GDP at PPP and it became a newly industrialised country and a major exporter in the 1990s. Manufacturing, agriculture, and tourism are leading sectors of the economy and it is considered a middle power in the region and around the world. The country has always been called Mueang Thai by its citizens, by outsiders prior to 1949, it was usually known by the exonym Siam. The word Siam has been identified with the Sanskrit Śyāma, the names Shan and A-hom seem to be variants of the same word. The word Śyâma is possibly not its origin, but a learned, another theory is the name derives from Chinese, Ayutthaya emerged as a dominant centre in the late fourteenth century. The Chinese called this region Xian, which the Portuguese converted into Siam, the signature of King Mongkut reads SPPM Mongkut King of the Siamese, giving the name Siam official status until 24 June 1939 when it was changed to Thailand. Thailand was renamed Siam from 1945 to 11 May 1949, after which it reverted to Thailand. According to George Cœdès, the word Thai means free man in the Thai language, ratcha Anachak Thai means kingdom of Thailand or kingdom of Thai. Etymologically, its components are, ratcha, -ana- -chak, the Thai National Anthem, written by Luang Saranupraphan during the extremely patriotic 1930s, refers to the Thai nation as, prathet Thai. The first line of the anthem is, prathet thai ruam lueat nuea chat chuea thai, Thailand is the unity of Thai flesh. There is evidence of habitation in Thailand that has been dated at 40,000 years before the present. Similar to other regions in Southeast Asia, Thailand was heavily influenced by the culture and religions of India, Thailand in its earliest days was under the rule of the Khmer Empire, which had strong Hindu roots, and the influence among Thais remains even today. Voretzsch believes that Buddhism must have been flowing into Siam from India in the time of the Indian Emperor Ashoka of the Maurya Empire, later Thailand was influenced by the south Indian Pallava dynasty and north Indian Gupta Empire. The Menam Basin was originally populated by the Mons, and the location of Dvaravati in the 7th century, the History of the Yuan mentions an embassy from the kingdom of Sukhothai in 1282

26.
Counting rods
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Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient China, Japan, Korea, and Vietnam. They are placed horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals and they are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period to the 16th century. Counting rods were used by ancient Chinese for more two thousand years. In 1954, forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chu Grave No.15 in Changsha, in 1973, archeologists unearthed a number of wood scripts from a Han dynasty tomb in Hubei. On one of the scripts was written, “当利二月定算”. This is one of the earliest examples of using counting rod numerals in writing, in 1976, a bundle of Western Han counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it, Laozi said a good calculator doesnt use counting rods, the Book of Han recorded, they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces. At first calculating rods were round in section, but by the time of the Sui dynasty triangular rods were used to represent positive numbers. After the abacus flourished, counting rods were abandoned except in Japan, counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used, generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc. while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that one is vertical, ten is horizontal, red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero, though they had no symbol for the latter, later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is important to understanding written transcription of rod numerals on manuscripts correctly. In the same manuscript,405 was transcribed as, with a space in between for obvious reasons, and could in no way be interpreted as 45. In other words, transcribed rod numerals may not be positional, the value of a number depends on its physical position on the counting board. A9 at the rightmost position on the stands for 9. Moving the batch of rods representing 9 to the one position gives 9 or 90

27.
China
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China, officially the Peoples Republic of China, is a unitary sovereign state in East Asia and the worlds most populous country, with a population of over 1.381 billion. The state is governed by the Communist Party of China and its capital is Beijing, the countrys major urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin and Hong Kong. China is a power and a major regional power within Asia. Chinas landscape is vast and diverse, ranging from forest steppes, the Himalaya, Karakoram, Pamir and Tian Shan mountain ranges separate China from much of South and Central Asia. The Yangtze and Yellow Rivers, the third and sixth longest in the world, respectively, Chinas coastline along the Pacific Ocean is 14,500 kilometers long and is bounded by the Bohai, Yellow, East China and South China seas. China emerged as one of the worlds earliest civilizations in the basin of the Yellow River in the North China Plain. For millennia, Chinas political system was based on hereditary monarchies known as dynasties, in 1912, the Republic of China replaced the last dynasty and ruled the Chinese mainland until 1949, when it was defeated by the communist Peoples Liberation Army in the Chinese Civil War. The Communist Party established the Peoples Republic of China in Beijing on 1 October 1949, both the ROC and PRC continue to claim to be the legitimate government of all China, though the latter has more recognition in the world and controls more territory. China had the largest economy in the world for much of the last two years, during which it has seen cycles of prosperity and decline. Since the introduction of reforms in 1978, China has become one of the worlds fastest-growing major economies. As of 2016, it is the worlds second-largest economy by nominal GDP, China is also the worlds largest exporter and second-largest importer of goods. China is a nuclear weapons state and has the worlds largest standing army. The PRC is a member of the United Nations, as it replaced the ROC as a permanent member of the U. N. Security Council in 1971. China is also a member of numerous formal and informal multilateral organizations, including the WTO, APEC, BRICS, the Shanghai Cooperation Organization, the BCIM, the English name China is first attested in Richard Edens 1555 translation of the 1516 journal of the Portuguese explorer Duarte Barbosa. The demonym, that is, the name for the people, Portuguese China is thought to derive from Persian Chīn, and perhaps ultimately from Sanskrit Cīna. Cīna was first used in early Hindu scripture, including the Mahābhārata, there are, however, other suggestions for the derivation of China. The official name of the state is the Peoples Republic of China. The shorter form is China Zhōngguó, from zhōng and guó and it was then applied to the area around Luoyi during the Eastern Zhou and then to Chinas Central Plain before being used as an occasional synonym for the state under the Qing

28.
Japan
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Japan is a sovereign island nation in Eastern Asia. Located in the Pacific Ocean, it lies off the eastern coast of the Asia Mainland and stretches from the Sea of Okhotsk in the north to the East China Sea, the kanji that make up Japans name mean sun origin. 日 can be read as ni and means sun while 本 can be read as hon, or pon, Japan is often referred to by the famous epithet Land of the Rising Sun in reference to its Japanese name. Japan is an archipelago consisting of about 6,852 islands. The four largest are Honshu, Hokkaido, Kyushu and Shikoku, the country is divided into 47 prefectures in eight regions. Hokkaido being the northernmost prefecture and Okinawa being the southernmost one, the population of 127 million is the worlds tenth largest. Japanese people make up 98. 5% of Japans total population, approximately 9.1 million people live in the city of Tokyo, the capital of Japan. Archaeological research indicates that Japan was inhabited as early as the Upper Paleolithic period, the first written mention of Japan is in Chinese history texts from the 1st century AD. Influence from other regions, mainly China, followed by periods of isolation, from the 12th century until 1868, Japan was ruled by successive feudal military shoguns who ruled in the name of the Emperor. Japan entered into a period of isolation in the early 17th century. The Second Sino-Japanese War of 1937 expanded into part of World War II in 1941, which came to an end in 1945 following the bombings of Hiroshima and Nagasaki. Japan is a member of the UN, the OECD, the G7, the G8, the country has the worlds third-largest economy by nominal GDP and the worlds fourth-largest economy by purchasing power parity. It is also the worlds fourth-largest exporter and fourth-largest importer, although Japan has officially renounced its right to declare war, it maintains a modern military with the worlds eighth-largest military budget, used for self-defense and peacekeeping roles. Japan is a country with a very high standard of living. Its population enjoys the highest life expectancy and the third lowest infant mortality rate in the world, in ancient China, Japan was called Wo 倭. It was mentioned in the third century Chinese historical text Records of the Three Kingdoms in the section for the Wei kingdom, Wa became disliked because it has the connotation of the character 矮, meaning dwarf. The 倭 kanji has been replaced with the homophone Wa, meaning harmony, the Japanese word for Japan is 日本, which is pronounced Nippon or Nihon and literally means the origin of the sun. The earliest record of the name Nihon appears in the Chinese historical records of the Tang dynasty, at the start of the seventh century, a delegation from Japan introduced their country as Nihon

29.
Chinese numerals
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Chinese numerals are words and characters used to denote numbers in Chinese. Today speakers of Chinese use three written numeral systems, the system of Arabic numerals used worldwide, and two indigenous systems, the more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared with languages of the Chinese cultural sphere such as Japanese, Korean. The other indigenous system is the Suzhou numerals, or huama, a positional system and these were once used by Chinese mathematicians, and later in Chinese markets, such as those in Hong Kong before the 1990s, but have been gradually supplanted by Arabic numerals. The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals, similar to spelling-out numbers in English, it is not an independent system per se. Since it reflects spoken language, it not use the positional system as in Arabic numerals. There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, there are two sets of characters for Chinese numerals, one for everyday writing and one for use in commercial or financial contexts known as dàxiě. A forger could easily change the everyday characters 三十 to 五千 just by adding a few strokes and that would not be possible when writing using the financial characters 參拾 and 伍仟. They are also referred to as bankers numerals, anti-fraud numerals, for the same reason, rod numerals were never used in commercial records. T denotes Traditional Chinese characters, S denotes Simplified Chinese characters, in the PLA, some numbers will have altered names when used for clearer radio communications. They are,0, renamed 洞 lit, hole 1, renamed 幺 lit. small 2, renamed 两 lit. Double 7, renamed 拐 lit. cane, kidnap, turn 9, hook For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan. To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, the ROC government in Taiwan uses 兆 to mean 1012 in official documents. Numerals beyond 載 zài come from Buddhist texts in Sanskrit, but are found in ancient texts. Some of the words are still being used today. The following are characters used to denote small order of magnitude in Chinese historically, with the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse. In the Peoples Republic of China, the translations for the SI prefixes in 1981 were different from those used today, the Republic of China defined 百萬 as the translation for mega

30.
Binary numeral system
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra

31.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three

32.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623, in 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and information theorist, for, among other reasons and he started developing this machine in 1834, and in less than two years, he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base

Computer science
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Ada Lovelace is credited with writing the first algorithm intended for processing on a computer.
Computer science
Computer science
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The German military used the Enigma machine (shown here) during World War II for communications they wanted kept secret. The large-scale decryption of Enigma traffic at Bletchley Park was an important factor that contributed to Allied victory in WWII.
Computer science
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Digital logic

33.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1

34.
Balanced ternary
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Balanced ternary is a non-standard positional numeral system, useful for comparison logic. While it is a number system, in the standard ternary system. The digits in the balanced ternary system have values −1,0, different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T represents −1, while 0 and 1 represent themselves, other conventions include using − and + to represent −1 and 1 respectively, or using Greek letter theta, which resembles a minus sign in a circle, to represent −1. In Setun printings, −1 is represented as overturned 1,1, the notation has a number of computational advantages over regular binary. Particularly, the plus–minus consistency cuts down the rate in multi-digit multiplication. Balanced ternary also has a number of advantages over traditional ternary. Particularly, the multiplication table has no carries in balanced ternary. A possible use of balanced ternary is to represent if a list of values in a list is less than, equal to or greater than the corresponding value in a second list. Balanced ternary can also represent all integers without using a separate minus sign, in the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3n. This is useful when converting between decimal and balanced ternary, in the following the strings denoting balanced ternary carry the suffix, bal3. For instance, −2/3dec = −1 + 1/3 = −1×30 + 1×3−1 = T. 1bal3, an integer is divisible by three if and only if the digit in the units place is zero. We may check the parity of a balanced ternary integer by checking the parity of the sum of all trits and this sum has the same parity as the integer itself. Balanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the radix point, in decimal or binary, integer values and terminating fractions have multiple representations. For example,110 =0.1 =0.10 =0.09, and,12 =0. 1bin =0. 10bin =0. 01bin. Some balanced ternary fractions have multiple representations too, for example,16 =0. 1Tbal3 =0. 01bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point, but, in balanced ternary, we cant omit the rightmost trailing infinite –1s after the radix point in order to gain a representations of integer or terminating fraction. Donald Knuth has pointed out that truncation and rounding are the operation in balanced ternary — they produce exactly the same result

35.
Setun
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Setun was a computer developed in 1958 at Moscow State University. It was built under the leadership of Sergei Sobolev and Nikolay Brusentsov and it was the most modern ternary computer, using the balanced ternary numeral system and three-valued ternary logic instead of the two-valued binary logic prevalent in other computers. The computer was built to fulfill the needs of Moscow State University and it was manufactured at the Kazan Mathematical plant. 50 computers were built from 1959 and production was halted in 1965, the characteristic operating memory consisted of 162 trits with additional 1944 trits on magnetic drum. Between 1965 and 1970, a binary computer was used at Moscow State University to replace it. Although this replacement binary computer performed equally well, it had 2.5 times the cost of the Setun, in 1970, a new model of the ternary computer, the Setun-70, was developed. Setun was named after the Setun River, which ends near Moscow University, DSSP is a programming language designed for Setun. It was created by students in the laboratory of Nikolay Brusentsov at the Computer Science department of the Moscow State University in 1980, the 32-bit version was created in 1989. DSSP is similar to the Forth programming language, both are examples of stack-based languages, the underlying ideology of DSSP was to reduce the semantic gaps between the human interface and the computer system. One principle was that there should only be one language to control, another was the principle of one word of text – one word of machine code. DSSPs structure stays very close to the actual machine and it uses reverse Polish notation, a stack-oriented form of calculation. The first document in English regarding this obscure language distinguishes DSSP from Forth in the following manner and that is why DSSP has not versions, but only extensions. But they are similar and it is a fact of great importance

36.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected

37.
Signed-digit representation
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In mathematical notation for numbers, signed-digit representation is a positional system with signed digits, the representation may not be unique. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries, in the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead. Challenges in calculation stimulated early authors Colson and Cauchy to use signed-digit representation, the further step of replacing negated digits with new ones was suggested by Selling and Cajori. In balanced form, the digits are drawn from a range −k to − k, for balanced forms, odd base numbers are advantageous. With an odd number, truncation and rounding become the same operation. A notable example is balanced ternary, where the base is b =3, balanced ternary uses the minimum number of digits in a balanced form. Balanced decimal uses digits from −5 to +4, balanced base nine, with digits from −4 to +4 provides the advantages of an odd-base balanced form with a similar number of digits, and is easy to convert to and from balanced ternary. Other notable examples include Booth encoding and non-adjacent form, both of which use a base of b =2, and both of which use numerals with the values −1,0, and +1, note that signed-digit representation is not necessarily unique. The oral and written forms of numbers in the Punjabi language use a form of a numeral one written as una or un. This negative one is used to form 19,29, …,89 from the root for 20,30, similarly, the Sesotho language utilizes negative numerals to form 8s and 9s. 8 robeli meaning break two i. e. two fingers down 9 robong meaning break one i. e. one finger down In 1928, Florian Cajori noted the theme of signed digits, starting with Colson. In his book History of Mathematical Notations, Cajori titled the section Negative numerals, eduard Selling advocated inverting the digits 1,2,3,4, and 5 to indicate the negative sign. He also suggested snie, jes, jerd, reff, most of the other early sources used a bar over a digit to indicate a negative sign for a it. For completeness, Colson uses examples and describes addition, multiplication and division using a table of multiples of the divisor and he explains the convenience of approximation by truncation in multiplication. Colson also devised an instrument that calculated using signed digits, Negative base Redundant binary representation J. P. Balantine A Digit for Negative One, American Mathematical Monthly 32,302. Augustin-Louis Cauchy Sur les moyens deviter les erreurs dans les calculs numerique, also found in Oevres completes Ser. Lui Han, Dongdong Chen, Seok-Bum Ko, Khan A. Wahid Non-speculative Decimal Signed Digit Adder from Department of Electrical and Computer Engineering, rudolf Mehmke Numerisches Rechen, §4 Beschränkung in den verwendeten Ziffern, Kleins encyclopedia, I-2, p.944

38.
Florian Cajori
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Florian Cajori was a Swiss-American historian of mathematics. Florian Cajori immigrated to the United States at the age of sixteen and he received both his bachelors and masters degrees from the University of Wisconsin–Madison. He taught for a few years at Tulane University, before being appointed as professor of applied mathematics there in 1887 and he was then driven north by tuberculosis. While in Colorado, he received his doctorate from Tulane in 1894, cajoris A History of Mathematics was the first popular presentation of the history of mathematics in the United States. He remained in Berkeley, California until his death in 1930, Cajori did no original mathematical research unrelated to the history of mathematics. In addition to his numerous books, he also contributed highly recognized and his last work was a revision of Andrew Mottes 1729 translation of Newtons Principia, vol.1 The Motion of Bodies, but he died before it was completed. The work was finished by R. T. Crawford of Berkeley,1893, A History of Mathematics, Macmillan & Company. 1898, A History of Elementary Mathematics, Macmillan,1909, A History of the Logarithmic Slide Rule and Allied Instruments The Engineering News Publishing Company. 1919, A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse,1920, On the History of Gunters Scale and the Slide Rule during the Seventeenth Century Vol.1, University of California Press. 1928, A History of Mathematical Notations The Open Court Company,1934, Sir Isaac Newtons Mathematical Principles of Natural Philosophy and His System of the World tr. Andrew Motte, rev. 1923, The History of Notations of the Calculus, Florian Cajori at the Mathematics Genealogy Project Florian Cajori. A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse

39.
Computer design
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In computer engineering, computer architecture is a set of rules and methods that describe the functionality, organization, and implementation of computer systems. Some definitions of architecture define it as describing the capabilities and programming model of a computer, in other definitions computer architecture involves instruction set architecture design, microarchitecture design, logic design, and implementation. The first documented computer architecture was in the correspondence between Charles Babbage and Ada Lovelace, describing the analytical engine, johnson had the opportunity to write a proprietary research communication about the Stretch, an IBM-developed supercomputer for Los Alamos National Laboratory. Brooks went on to develop the IBM System/360 line of computers. Later, computer users came to use the term in many less-explicit ways, the earliest computer architectures were designed on paper and then directly built into the final hardware form. The discipline of architecture has three main subcategories, Instruction Set Architecture, or ISA. The ISA defines the code that a processor reads and acts upon as well as the word size, memory address modes, processor registers. Microarchitecture, or computer organization describes how a processor will implement the ISA. The size of a computers CPU cache for instance, is an issue that generally has nothing to do with the ISA, system Design includes all of the other hardware components within a computing system. These include, Data processing other than the CPU, such as memory access Other issues such as virtualization, multiprocessing. There are other types of computer architecture, E. g. the C, C++, or Java standards define different Programmer Visible Macroarchitecture. UISA —a group of machines with different hardware level microarchitectures may share a common microcode architecture, pin Architecture, The hardware functions that a microprocessor should provide to a hardware platform, e. g. the x86 pins A20M, FERR/IGNNE or FLUSH. Also, messages that the processor should emit so that external caches can be invalidated, pin architecture functions are more flexible than ISA functions because external hardware can adapt to new encodings, or change from a pin to a message. The term architecture fits, because the functions must be provided for compatible systems, the purpose is to design a computer that maximizes performance while keeping power consumption in check, costs low relative to the amount of expected performance, and is also very reliable. For this, many aspects are to be considered which includes Instruction Set Design, Functional Organization, Logic Design, the implementation involves Integrated Circuit Design, Packaging, Power, and Cooling. Optimization of the design requires familiarity with Compilers, Operating Systems to Logic Design, an instruction set architecture is the interface between the computers software and hardware and also can be viewed as the programmers view of the machine. Computers do not understand high level languages such as Java, C++, a processor only understands instructions encoded in some numerical fashion, usually as binary numbers. Software tools, such as compilers, translate those high level languages into instructions that the processor can understand, besides instructions, the ISA defines items in the computer that are available to a program—e. g

40.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

41.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw

42.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.

43.
History of writing ancient numbers
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The first numbers were invented in India. The first method of counting was counting on fingers and this evolved into sign language for the hand-to-eye communication of numbers which, while not writing, gave way to written numbers. Tallies made by carving notches in wood, bone, and stone were used for at least forty thousand years, stone age cultures, including ancient Native American groups, used tallies for gambling with horses, slaves, personal services and trade-goods. Roman numerals evolved from this system of cutting notches. It was once believed that they came from alphabetic symbols or from pictographs, the earliest known writing for record keeping evolved from a system of counting using small clay tokens. The earliest tokens now known are those two sites in the Zagros region of Iran, Tepe Asiab and Ganj-i-Dareh Tepe. To create a record that represented two sheep, they selected two round clay tokens each having a + sign baked into it, there was a token for one sheep, a different token for ten sheep, a different token for ten goats, etc. Thirty-two sheep would be represented by three ten-sheep tokens followed on the string by two one-sheep tokens. To ensure that nobody could alter the number and type of tokens, they invented a clay envelope shaped like a ball into which the tokens on a string were placed, sealed. If anybody disputed the number, they could open the clay envelope. Since there was any need to break open the envelope. An alternative method was to seal the knot in each string of tokens with a solid oblong bulla of clay having impressed symbols, while the string of tokens dangled outside of the bulla. Beginning about 3500 BC the tokens and envelopes were replaced by numerals impressed with a round stylus at different angles in clay tablets which were then baked. A sharp stylus was used to carve pictographs representing various tokens, each sign represented both the commodity being counted and the quantity or volume of that commodity. Abstract numerals, dissociated from the thing being counted, were invented about 3100 BC, the things being counted were indicated by pictographs carved with a sharp stylus next to round-stylus numerals. The Sumerians had an assortment of incompatible number systems. For instance, at about 3100 BC in the city of Uruk, in this city, there were separate number systems for counting discrete objects, cheese and grain products, volumes of grain, beer ingredients, weights, land areas, and time and calendar units. Furthermore, these changed over time, for instance, numbers for counting volumes of grain changed when the size of the baskets changed

History of writing ancient numbers
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Numeral systems
History of writing ancient numbers
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Middle Babylonian legal tablet from Alalah in its envelope
History of writing ancient numbers
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Uruk period: globular envelope with a cluster of accountancy tokens, from Susa. Louvre Museum

44.
Tally marks
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Tally marks, also called hash marks, are a unary numeral system. They are a form of used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, however, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were historically used for this purpose. Counting aids other than body parts appear in the Upper Paleolithic, the oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africas Late Stone Age. The so-called Wolf bone is an artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia. Dated to the Aurignacian, approximately 30,000 years ago, the head of an ivory Venus figurine was excavated close to the bone. The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old, upon discovery, it was thought to portray a series of prime numbers. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10. Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a lunar calendar. Tally marks are typically clustered in groups of five for legibility, roman numerals, the Chinese numerals for one through three, and rod numerals were derived from tally marks, as possibly was the ogham script

Tally marks
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Counting using tally marks at Hanakapiai Beach. The number shown is 82.
Tally marks
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In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.

45.
Indigenous peoples of the Americas
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The indigenous peoples of the Americas are the pre-Columbian peoples of the Americas and their descendants. The term Amerindian is used in Quebec, the Guianas, Indigenous peoples of the United States are commonly known as Native Americans or American Indians, and Alaska Natives. Application of the term Indian originated with Christopher Columbus, who, in his search for Asia, eventually, the Americas came to be known as the West Indies, a name still used to refer to the islands of the Caribbean Sea. This led to the blanket term Indies and Indians for the indigenous inhabitants, although some indigenous peoples of the Americas were traditionally hunter-gatherers—and many, especially in the Amazon basin, still are—many groups practiced aquaculture and agriculture. The impact of their agricultural endowment to the world is a testament to their time, although some societies depended heavily on agriculture, others practiced a mix of farming, hunting, and gathering. In some regions the indigenous peoples created monumental architecture, large-scale organized cities, chiefdoms, states, and empires. Many parts of the Americas are still populated by peoples, some countries have sizable populations, especially Belize, Bolivia, Chile, Ecuador, Greenland, Guatemala, Mexico. At least a different indigenous languages are spoken in the Americas. Some, such as the Quechuan languages, Aymara, Guaraní, Mayan languages, many also maintain aspects of indigenous cultural practices to varying degrees, including religion, social organization, and subsistence practices. Like most cultures, over time, cultures specific to many indigenous peoples have evolved to incorporate traditional aspects, some indigenous peoples still live in relative isolation from Western culture and a few are still counted as uncontacted peoples. The specifics of Paleo-Indian migration to and throughout the Americas, including the dates and routes traveled, are the subject of ongoing research. According to archaeological and genetic evidence, North and South America were the last continents in the world with human habitation. During the Wisconsin glaciation, 50–17,000 years ago, falling sea levels allowed people to move across the bridge of Beringia that joined Siberia to northwest North America. Alaska was a glacial refugium because it had low snowfall, allowing a small population to exist, the Laurentide Ice Sheet covered most of North America, blocking nomadic inhabitants and confining them to Alaska for thousands of years. Indigenous genetic studies suggest that the first inhabitants of the Americas share a single population, one that developed in isolation. The isolation of these peoples in Beringia might have lasted 10–20,000 years, around 16,500 years ago, the glaciers began melting, allowing people to move south and east into Canada and beyond. These people are believed to have followed herds of now-extinct Pleistocene megafauna along ice-free corridors that stretched between the Laurentide and Cordilleran Ice Sheets. Another route proposed involves migration - either on foot or using primitive boats - along the Pacific Northwest coast to the south, archeological evidence of the latter would have been covered by the sea level rise of more than 120 meters since the last ice age

46.
Sumer
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Living along the valleys of the Tigris and Euphrates, Sumerian farmers were able to grow an abundance of grain and other crops, the surplus of which enabled them to settle in one place. Proto-writing in the dates back to c.3000 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr and date back to 3300 BC, modern historians have suggested that Sumer was first permanently settled between c.5500 and 4000 BC by a West Asian people who spoke the Sumerian language, an agglutinative language isolate. These conjectured, prehistoric people are now called proto-Euphrateans or Ubaidians, some scholars contest the idea of a Proto-Euphratean language or one substrate language. Reliable historical records begin much later, there are none in Sumer of any kind that have dated before Enmebaragesi. Juris Zarins believes the Sumerians lived along the coast of Eastern Arabia, todays Persian Gulf region, Sumerian civilization took form in the Uruk period, continuing into the Jemdet Nasr and Early Dynastic periods. During the 3rd millennium BC, a cultural symbiosis developed between the Sumerians, who spoke a language isolate, and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the 3rd millennium BC as a Sprachbund, Sumer was conquered by the Semitic-speaking kings of the Akkadian Empire around 2270 BC, but Sumerian continued as a sacred language. Native Sumerian rule re-emerged for about a century in the Neo-Sumerian Empire or Third Dynasty of Ur approximately 2100-2000 BC, the term Sumerian is the common name given to the ancient non-Semitic-speaking inhabitants of Mesopotamia, Sumer, by the East Semitic-speaking Akkadians. The Sumerians referred to themselves as ùĝ saĝ gíg ga, phonetically /uŋ saŋ gi ga/, literally meaning the black-headed people, the Akkadian word Shumer may represent the geographical name in dialect, but the phonological development leading to the Akkadian term šumerû is uncertain. Hebrew Shinar, Egyptian Sngr, and Hittite Šanhar, all referring to southern Mesopotamia, in the late 4th millennium BC, Sumer was divided into many independent city-states, which were divided by canals and boundary stones. Each was centered on a dedicated to the particular patron god or goddess of the city. The Sumerian city-states rose to power during the prehistoric Ubaid and Uruk periods, classical Sumer ends with the rise of the Akkadian Empire in the 23rd century BC. Following the Gutian period, there is a brief Sumerian Renaissance in the 21st century BC, the Amorite dynasty of Isin persisted until c.1700 BC, when Mesopotamia was united under Babylonian rule. The Sumerians were eventually absorbed into the Akkadian population, 2500–2334 BC Akkadian Empire period, c. 2218–2047 BC Ur III period, c, 2047–1940 BC The Ubaid period is marked by a distinctive style of fine quality painted pottery which spread throughout Mesopotamia and the Persian Gulf. It appears that this culture was derived from the Samarran culture from northern Mesopotamia and it is not known whether or not these were the actual Sumerians who are identified with the later Uruk culture

47.
Mixed radix
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Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the smaller one. 32,5,7,45,15,500. ∞,7,24,60,60,1000 or as 32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its base attached as a subscript. The base for each digit is the number of corresponding units that make up the larger unit. As a consequence there is no base for the first digit, the most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks, one variant uses tridecimal months, quaternary weeks, and septenary days. Time is further divided by quadrivigesimal hours, sexagesimal minutes and seconds, a mixed radix numeral system can often benefit from a tabular summary. m. On Wednesday, and 070201202602460 would be 12,02,24 a. m. on Sunday, ad hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices, a short count tzolkin overlaps vigesimal named days with tridecimal numbered days. A haab consists of vigesimal days, octodecimal months, and base-52 years forming a round, in addition, a long count of vigesimal days, octodecimal winal, then vigesimal tun, katun, baktun, etc. tracks historical dates. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. APL and J include operators to convert to and from mixed-radix systems, another proposal is the so-called factorial number system, For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal, 5×5. It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one, −1 There is a natural mapping between the integers 0. N. −1 and permutations of n elements in lexicographic order, the above equation is a particular case of the following general rule for any radix base representation which expresses the fact that any radix base representation is unambiguous and complete. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Über einfache Zahlensysteme, Zeitschrift für Math. Mixed Radix Calculator — Mixed Radix Calculator in C#

48.
Time
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Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers, one view is that time is part of the fundamental structure of the universe—a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is referred to as Newtonian time. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, Time in physics is unambiguously operationally defined as what a clock reads. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities, Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition. The operational definition leaves aside the question there is something called time, apart from the counting activity just mentioned, that flows. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be there is a subjective component to time. Temporal measurement has occupied scientists and technologists, and was a motivation in navigation. Periodic events and periodic motion have long served as standards for units of time, examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is also of significant social importance, having economic value as well as value, due to an awareness of the limited time in each day. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day, increasingly, personal electronic devices display both calendars and clocks simultaneously. The number that marks the occurrence of an event as to hour or date is obtained by counting from a fiducial epoch—a central reference point. Artifacts from the Paleolithic suggest that the moon was used to time as early as 6,000 years ago. Lunar calendars were among the first to appear, either 12 or 13 lunar months, without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months

49.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles

Angle
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An angle enclosed by rays emanating from a vertex.

50.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2